Question: Wang Lei and Amira were asked to find an explicit formula for the sequence $30\,,\,150\,,\,750\,,\,3750,...$, where the first term should be $g(1)$. Wang Lei said the formula is $g(n)=30\cdot5^{{n-1}}$, and Amira said the formula is $g(n)=6\cdot5^{{n}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Wang Lei (Choice B) B Only Amira (Choice C) C Both Wang Lei and Amira (Choice D) D Neither Wang Lei nor Amira
Answer: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{3750}{750}=\dfrac{750}{150}=\dfrac{150}{30}={5}$ We see that the constant ratio between successive terms is ${5}$. In other words, we can find any term by starting with the first term and multiplying by ${5}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $g(n)$ ${30}\cdot\!{5}^{0}$ ${30}\cdot\!{5}^{1}$ ${30}\cdot\!{5}^{2}$ ${30}\cdot\!{5}^{3}$ We can see that every term is the product of the first term, ${30}$, and a power of the constant ratio, ${5}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (note that ${30}$ is the first term and ${5}$ is the constant ratio): $g(n)={30}\cdot{5}^{{\,n-1}}$ So Wang Lei is definitely right. What about Amira? We can see that in Amira's formula, the constant ratio is taken to the $n^{\text{th}}$ power. Let's expand the power in Wang Lei's formula to arrive at a similar expression form: $\begin{aligned} g(n)= &{30}\cdot{5}^{{\,n-1}}\\\\ = & 30\cdot5^{{n}}\cdot5^{{-1}}\\\\ = & 30\cdot \left(\dfrac{1}{5}\right) \cdot 5^{{n}}\\\\ = &6\cdot5^{{n}}\end{aligned}$ We obtained Amira's formula, which means it's also a correct explicit formula for $g(n)$. Both Wang Lei and Amira got a correct explicit formula.